What books are good to learn the math in intro physics?

[astroman707]

I'm struggling with the math used in my college's calc-based honors physics class, even though I've taken calculus 1.
---What are some good books/resources to learn the math used in introductory physics?---

Preferably, it'd be nice if the math was taught using examples in physics. Having that context when learning the math would be really helpful in understanding it's purpose in my courses. I've been recommended mathematics for the physical sciences by Mary Boas, but most of the math in that book seemed to be for after introductory physics, and had no clear applications/context for me.

 

[symbolipoint]

If you intend for the Mathematics needed for the typical beginning Physics for science and engineering students, then the necessary textbooks of Mathematics are what your school would assign for Calculus & Analytic Geometry 1, 2, 3, and your combination course Introductory Linear Algebra & Differential Equations. The exercises and application exercises in these Mathematics books should be more than sufficient to prepare you for the calculus-based Physics series of courses. Most of the conditioning you need for studying Physics, unfortunately, MUST come with your Physics courses. You need some flexibility and not try to look for ways to make yourself too narrow just for Physics. From my own experience, most of the harder and Calculus mathematics was used in the Electricity & Magnetism course of the sci & engng Physics series.

 

[symbolipoint]

I just reread part of what you wrote in #1.
If you are struggling and in just Calculus 1, you need to review something. I can not say exactly what it is. Maybe review you intermediate or college algebra. Maybe review Trigonometry; maybe review thoroughly all of Calculus 1, even if it takes 2 or 3 months. Be sure you are studying enough and doing more practice exercises than are assigned. Some people need more time than just a single semester.
"Honors Physics" based on Calc 1 ? You are likely struggling to adjust to how to solve Physics problems, and this is what MANY students go through. REVIEW! REVIEW MORE. If you pass your current course, review it again on your own after finishing the course.

 

[verty]

I recommend this blue book partnered with https://ocw.mit.edu/courses/mathematics/18-01-single-variable-calculus-fall-2006/video-lectures/ to get you out of your schtuck. I think you need solidly good teaching to get past whatever your issue is.

 

[Dr. Courtney]

Depends on the course.

Most of the troubles I've seen are with Algebra 1. Take Algebra 1 in ALEKS.

 

[symbolipoint]

Depends on the course.

Most of the troubles I've seen are with Algebra 1. Take Algebra 1 in ALEKS.

Interesting that this is still the common belief. One would think that by the time students start Physics 1, that they have already done one semester of Calculus 1 and that "Algebra 1" should have become a basic part of the students' skills and knowledge, and that learning to handle Physics 1 becomes the current chore, and that the students have gained enough understanding - because of the reenforcement gained since Algebra 1, reaching into Calculus 1 and other courses and experiences requiring Algebra skills. Honestly, course repetition also helps with Algebra, which was my case. With me, basic Algebra 1&2 were not my trouble; but learning to handle them with Physics was. This too, made me better at using basic algebra.

 

[Wrichik Basu]

I really find myself in trouble when I find people mentioning "Calculus 1" or "Algebra 1". Not being from USA, I am not quite accustomed to these terms. In fact, I study from online lectures where these terns are not used. So, even if they are in use in Indian universities, I am not aware of them.

Anyways, if I assume from post #6 that "Algebra 1" is Linear Algebra, then you can have a look at these courses:

Mathematics methods in Physics - 1 by Prof. Samudra Roy

Linear Algebra by Prof. K. C. Sivakumar (this is a rigorous course in maths)

For a very basic course on calculus, you can refer to:

Basic Calculus for Engineers, Scientists and Economists by Dr. Joydeep Dutta

For advanced courses, you can yourself perform a search in the NPTEL website. There are a lot of results when you type in "Calculus".

Another course on basic maths for Physicists is:

Selected Topics in Mathematical Physics by Prof. V. Balakrishnan

Hope that helps.

Wrichik.

 

[Dr. Courtney]

Interesting that this is still the common belief. One would think that by the time students start Physics 1, that they have already done one semester of Calculus 1 and that "Algebra 1" should have become a basic part of the students' skills and knowledge, and that learning to handle Physics 1 becomes the current chore, and that the students have gained enough understanding - because of the reenforcement gained since Algebra 1, reaching into Calculus 1 and other courses and experiences requiring Algebra skills. Honestly, course repetition also helps with Algebra, which was my case. With me, basic Algebra 1&2 were not my trouble; but learning to handle them with Physics was. This too, made me better at using basic algebra.

This has not been my experience. Perhaps it is because of several factors:
1. The Calculus courses at a couple places where I taught did a poor job reinforcing algebra skills. Lots of students at the community college and lower tier NC university reached Calculus and Physics with poor algebra skills, and most of the teachers were unwilling to make up the ground so they designed the courses so students could pass with weak basic algebra skills. Assuming a student who has passed Calculus is strong in algebra is faulty.
2. Even in "Calc-based" Physics courses that are heavily focused on problem solving, review of homework and tests tends to show that less than 25% of the problems require calculus, but they all require algebra 1 and about 40% require trig. This was certainly true of the Calc-based physics courses at West Point and USAFA, and I've also seen it to be true of many Physics courses for Engineering majors. My experience is that only the Physics courses for physics majors tend to require Calculus on more than 50% of assigned homework and test problems.

I really find myself in trouble when I find people mentioning "Calculus 1" or "Algebra 1". Not being from USA, I am not quite accustomed to these terms. In fact, I study from online lectures where these terns are not used. So, even if they are in use in Indian universities, I am not aware of them.

Anyways, if I assume from post #6 that "Algebra 1" is Linear Algebra, then you can have a look at these courses:

Algebra 1 is usually the first high school Algebra course. It is the course in basic algebra needed for all later math and physics that requires algebra.

 

[Wrichik Basu]

Algebra 1 is usually the first high school Algebra course. It is the course in basic algebra needed for all later math and physics that requires algebra.

Thanks for the explanation.

 

[astroman707]

Thanks for all the feedback. My issue is certainly not in the algebra, as that’s how I navigate all of physics. My issue is with understanding vectors on the level required of physics 1 & 2; and the relationships between physics concepts that are tied together with calculus 1-3, differential equations, etc.
I think a good point is made if I say that I can do all of the problems in the homework and exams, I understand conceptually why certain physics phenomena occur, but I cannot replicate the mathematical explananations my professor gives when introducing topics. For instance, I don’t know how to show how a physical pendulum works, how angular momentum and torque are related to each other, etc. I’ve been told that it’s important that I can replicate the “proofs”/relationships that my professor writes on the board, and understand the mathematical relationships between them.

 

[MidgetDwarf]

Hmmm.

A nice book that I learned calculus from was Thomas: Calculus with Analytical Geometry 3rd ed. It has very clear explanations. Diagrams are few but explained really well. Nice explanation of volumes, Pappus Theorems, derivatives, integrals, mean-value theorem etc. Teaches how to apply the calculus.
https://www.amazon.com/dp/B00GMPZBGA/?tag=pfamazon01-20

Book is 8 bucks with shipping.

https://www.amazon.com/gp/product/B002IA3LGU/?tag=pfamazon01-20
For differential equations. Very well explained explanations. I did not like the the explanation of the Laplace Transform. For an easy explanation of the laplace transform, look for Zill:Differential Equations with Boundary Problems. Any old edition of Zill will do.

This is my favorite ODE book. Very clear!

For linear algebra. Maybe look at Anton: Linear Algebra. But really, any introductory book on physics has sections teaching the linear algebra you need to know. I am certain you do not need to know about linear operators, vector spaces, etc at this point...

Look at the book series titled Alonso and Finn: Fundamental University Physics.
Very heavy use of calculus. Explanations are clear and concise. The whole book series can be expensive, but it is worth having. If you can get the set of 3 books for under 300, then do it! I still reference them, and they are a joy to read. It connected a lot of the dots for me. The author was amazing at connecting the physics ideas. Almost everything presented is derived.

Here is the first book:
https://www.amazon.com/dp/B0006BNQDG/?tag=pfamazon01-20

 

[Cryo]

Not sure what college is...

My go to book for standard maths toolset is Arfken & Weber "Mathematical Methods for Physicists" (https://www.amazon.com/dp/0123846544/?tag=pfamazon01-20)

Other favourites are:

LinAlg: Sheldon Axler "Linear algbera done right"
Tensors etc: Rund & Lovelock "Tensors, Differential Forms, and Variational Principles"
etc.

 

[jtbell]

Not sure what college is...

In the US, "college" is a synonym for undergraduate-level university studies. "Physics 1 & 2" here refers to an introductory calculus-based physics course using a textbook such as Halliday/Resnick/Walker's "Fundamentals of Physics." Some students study this in high school as "AP [Advanced Placement] Physics C". I understand that in some countries it's more common for students to do it in high school.

Arfken & Weber et al. are far above the mathematical level needed for this course.

 

[Cryo]

In the US, "college" is a synonym for undergraduate-level university studies. "Physics 1 & 2" here refers to an introductory calculus-based physics course using a textbook such as Halliday/Resnick/Walker's "Fundamentals of Physics." Some students study this in high school as "AP [Advanced Placement] Physics C". I understand that in some countries it's more common for students to do it in high school.

Arfken & Weber et al. are far above the mathematical level needed for this course.

I had my suspicions when I did not see any familiar books. Thanks for putting it right.

 

[Yggdresil]

Thanks for all the feedback. My issue is certainly not in the algebra, as that’s how I navigate all of physics. My issue is with understanding vectors on the level required of physics 1 & 2; and the relationships between physics concepts that are tied together with calculus 1-3, differential equations, etc.
I think a good point is made if I say that I can do all of the problems in the homework and exams, I understand conceptually why certain physics phenomena occur, but I cannot replicate the mathematical explananations my professor gives when introducing topics. For instance, I don’t know how to show how a physical pendulum works, how angular momentum and torque are related to each other, etc. I’ve been told that it’s important that I can replicate the “proofs”/relationships that my professor writes on the board, and understand the mathematical relationships between them.

Other than vectors none of this sounds like a math problem. Any introductory mechanics text should have enough information on vectors to avoid the need to consult a math linear algebra textbook. Then you just need to practice. For the rest of it, it certainly sounds like a failure to understand the physical concepts behind the math, in which case you’d be better served by rereading the sections in your physics text that you’re struggling with and practicing how to derive the equations from the first principles you’re given.